The crack in the blade is the most common type of fatigue damage for Francis turbines. However, the crack sometimes is difficult to be detected in time using the current monitoring system even when the crack is very large. To better monitor the crack, it is imperative to research the effect of a crack on the dynamic behavior of a Francis turbine. In this paper, the dynamic behavior of a Francis turbine runner model with a crack was researched numerically. The intact numerical model was first validated by the experimental data available. Then, a crack was created at the intersection line between one blade and the crown. The change in dynamic behavior with increasing crack length was investigated. Crack-induced vibration localization theory was used to explain the dynamic behavior changes due to the crack. Modal analysis showed that the adopted theory could basically explain the modal behavior change due to the crack. The FFT results of the modal shapes and the localization factors (LF) were used to explain the forced response changes due to the crack. Based on the above analysis, the challenge of crack monitoring was analyzed. This research can also provide some references for more advanced monitoring technologies.

In order to improve the performance of the draft tube in hydraulic turbine, a multi–objective optimization method for the draft tube is developed by combining the design of experiment (DOE), the radial basis function (RBF) and the non–dominated sorting genetic algorithm (NSGA–II) in this paper. The geometrical design variables of the median section in the draft tube and the cross section in its exit diffuser are considered as design parameters in this optimization, which objective function is to maximize the pressure recovery factor (Cp) and minimize the energy loss coefficient (ζ). The limited numbers of design matrix required for the shape optimization of the draft tube is generated by optimal Latin hypercube (OLH) method of the DOE technique, of which performances are evaluated through computational fluid dynamic (CFD) numerical simulation. For reducing of the computational consumption, the approximate model is used based on the RBF. The Pareto optimal solutions are finally performed using the NSGA–II for obtaining the best geometrical parameters of the draft tube. The optimization result of the draft tube shows a marked performance improvement over the original, which verifies the theoretical validity and feasibility of the proposed method in this paper.

In the development of a digital twin (DT) for hydropower turbines, dynamic modeling of the system (e.g., penstock, turbine, speed control) is crucial, along with all the necessary data interface, virtualization, and dashboard designs. Since the DT must mimic the actual dynamics of the hydropower turbine accurately, adaptive learning is required to train these dynamic models online so that the models in the DT can effectively follow the representation of the actual hydropower turbine dynamics accurately and reliably. This study presents an adaptive learning method for obtaining the hydropower turbine models for DT development of hydropower systems using recursive least squares algorithm. To simplify the formulation, the hydropower turbine under consideration was assumed to operate near a fixed operating point, where the system dynamics can be well represented by a set of linear differential equations with constant parameters. In this context, the well-known six-coefficient model for the Francis turbine was formulated as the starting point to obtain input and output models for the turbine. Then, an adaptive learning mechanism was developed to learn model parameters using the real-time data from a hydropower turbine testing system. This led to a semi-physical modeling in which the first principles and data-driven modeling are integrated to produce dynamic models for the DT development. Applications to a pilot system at the Norwegian University of Science and Technology (NTNU) were made, and the models learned adaptively using the data collected from the university’s pilot system. Desired modeling and validation results were obtained.